WEBVTT

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Which cell belongs to these four input bits?

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A repository explicit C M stores the answers to a rule with inputs A, B,

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C and D in an ordered grid. We will declare which inputs select rows and columns,

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split an assignment into those groups, and turn the binary groups into a checked matrix address.

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Our rule asks whether two checks disagree. The first
check is A and B: both must be true. The second is C or D:

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at least one must be true. Exclusive-or returns one when exactly one of these two checks passes.

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Use A and B for rows, and C and D for columns. The repository orders each group as zero-zero,

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zero-one, one-zero, one-one. The first bit is the more significant bit,

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so the corresponding numeric indices are zero, one, two, three.

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Take assignment one-zero-one-one in A, B, C, D order. Split it according to our declared groups.

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A B is one-zero, and C D is one-one.

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These groups identify a row and a column; they are not yet the output.

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For a two-bit binary group, the first position has weight two and the second has weight one.

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Row bits one-zero mean one times two plus zero times one.

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That gives row index two, the third displayed row.

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Column bits one-one mean one times two plus one times one.

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That gives column index three, the fourth displayed column.

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Follow the selected row and column to their intersection.
We have found the location for this assignment.

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Now evaluate the rule. A and B is one and zero, giving zero. C or D is one or one, giving one.

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The two checks differ, so exclusive-or returns one.
That is the value stored at row two, column three.

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Try one-one-one-zero. Split the bits into the same
two groups, find their indices, and decide the output.

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Be careful: the selected column contains a one bit, but that
does not by itself determine whether the whole rule is true.

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A B is one-one, so choose row three. C D is one-zero, so choose column two.

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Both component checks return one. They agree, so exclusive-or returns zero.

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The cell at row three, column two confirms that answer.

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One more example: zero-one-zero-zero. The row group gives index one,

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and the column group gives index zero. A and B is false, and C or D is false.

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Equal results again give zero. Exclusive-or rejects both agreement cases.

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The method is always the same: read the declared
variable order, split the assignment, convert each group,

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and select the intersection. Evaluating the original rule gives an independent check.

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Next we will change the split while keeping every answer the same.